Properties

Label 1445.2.d.g.866.4
Level 14451445
Weight 22
Character 1445.866
Analytic conductor 11.53811.538
Analytic rank 00
Dimension 1212
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1445,2,Mod(866,1445)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1445, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1445.866");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1445=5172 1445 = 5 \cdot 17^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1445.d (of order 22, degree 11, not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 11.538383092111.5383830921
Analytic rank: 00
Dimension: 1212
Coefficient field: Q[x]/(x12+)\mathbb{Q}[x]/(x^{12} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x12+18x10+83x8+152x6+111x4+22x2+1 x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 24 2^{4}
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 866.4
Root 1.52346i-1.52346i of defining polynomial
Character χ\chi == 1445.866
Dual form 1445.2.d.g.866.3

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q2.07061q2+2.52346iq3+2.28744q4+1.00000iq55.22511iq60.368961iq70.595174q83.36786q92.07061iq10+2.49810iq11+5.77226iq12+4.68778q13+0.763976iq142.52346q153.34250q16+6.97354q18+7.16938q19+2.28744iq20+0.931060q215.17260iq226.70055iq231.50190iq241.00000q259.70657q260.928289iq270.843976iq28+6.78165iq29+5.22511q30+4.81801iq31+8.11138q326.30387q33+0.368961q357.70378q361.93935iq3714.8450q38+11.8294iq390.595174iq40+2.35972iq411.92786q42+11.7105q43+5.71425iq443.36786iq45+13.8743iq46+1.65317q478.43468iq48+6.86387q49+2.07061q50+10.7230q52+6.81536q53+1.92213iq542.49810q55+0.219596iq56+18.0917iq5714.0422iq58+0.484372q595.77226q60+6.88307iq619.97624iq62+1.24261iq6310.1105q64+4.68778iq65+13.0529q661.87478q67+16.9086q690.763976q70+1.71958iq71+2.00446q720.286310iq73+4.01564iq742.52346iq75+16.3995q76+0.921703q7724.4942iq785.38563iq793.34250iq807.76109q814.88608iq829.94985q83+2.12974q8424.2479q8617.1132q871.48680iq88+4.30781q89+6.97354iq901.72961iq9115.3271iq9212.1581q933.42308q94+7.16938iq95+20.4688iq9612.7460iq9714.2124q988.41327iq99+O(q100)q-2.07061 q^{2} +2.52346i q^{3} +2.28744 q^{4} +1.00000i q^{5} -5.22511i q^{6} -0.368961i q^{7} -0.595174 q^{8} -3.36786 q^{9} -2.07061i q^{10} +2.49810i q^{11} +5.77226i q^{12} +4.68778 q^{13} +0.763976i q^{14} -2.52346 q^{15} -3.34250 q^{16} +6.97354 q^{18} +7.16938 q^{19} +2.28744i q^{20} +0.931060 q^{21} -5.17260i q^{22} -6.70055i q^{23} -1.50190i q^{24} -1.00000 q^{25} -9.70657 q^{26} -0.928289i q^{27} -0.843976i q^{28} +6.78165i q^{29} +5.22511 q^{30} +4.81801i q^{31} +8.11138 q^{32} -6.30387 q^{33} +0.368961 q^{35} -7.70378 q^{36} -1.93935i q^{37} -14.8450 q^{38} +11.8294i q^{39} -0.595174i q^{40} +2.35972i q^{41} -1.92786 q^{42} +11.7105 q^{43} +5.71425i q^{44} -3.36786i q^{45} +13.8743i q^{46} +1.65317 q^{47} -8.43468i q^{48} +6.86387 q^{49} +2.07061 q^{50} +10.7230 q^{52} +6.81536 q^{53} +1.92213i q^{54} -2.49810 q^{55} +0.219596i q^{56} +18.0917i q^{57} -14.0422i q^{58} +0.484372 q^{59} -5.77226 q^{60} +6.88307i q^{61} -9.97624i q^{62} +1.24261i q^{63} -10.1105 q^{64} +4.68778i q^{65} +13.0529 q^{66} -1.87478 q^{67} +16.9086 q^{69} -0.763976 q^{70} +1.71958i q^{71} +2.00446 q^{72} -0.286310i q^{73} +4.01564i q^{74} -2.52346i q^{75} +16.3995 q^{76} +0.921703 q^{77} -24.4942i q^{78} -5.38563i q^{79} -3.34250i q^{80} -7.76109 q^{81} -4.88608i q^{82} -9.94985 q^{83} +2.12974 q^{84} -24.2479 q^{86} -17.1132 q^{87} -1.48680i q^{88} +4.30781 q^{89} +6.97354i q^{90} -1.72961i q^{91} -15.3271i q^{92} -12.1581 q^{93} -3.42308 q^{94} +7.16938i q^{95} +20.4688i q^{96} -12.7460i q^{97} -14.2124 q^{98} -8.41327i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q4q2+12q412q84q98q15+4q16+28q18+24q19+16q2112q25+24q26+8q30+12q3216q33+16q35+20q3624q38+16q42+44q98+O(q100) 12 q - 4 q^{2} + 12 q^{4} - 12 q^{8} - 4 q^{9} - 8 q^{15} + 4 q^{16} + 28 q^{18} + 24 q^{19} + 16 q^{21} - 12 q^{25} + 24 q^{26} + 8 q^{30} + 12 q^{32} - 16 q^{33} + 16 q^{35} + 20 q^{36} - 24 q^{38} + 16 q^{42}+ \cdots - 44 q^{98}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1445Z)×\left(\mathbb{Z}/1445\mathbb{Z}\right)^\times.

nn 581581 11571157
χ(n)\chi(n) 1-1 11

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 −2.07061 −1.46414 −0.732072 0.681227i 0.761447π-0.761447\pi
−0.732072 + 0.681227i 0.761447π0.761447\pi
33 2.52346i 1.45692i 0.685087 + 0.728461i 0.259764π0.259764\pi
−0.685087 + 0.728461i 0.740236π0.740236\pi
44 2.28744 1.14372
55 1.00000i 0.447214i
66 − 5.22511i − 2.13314i
77 − 0.368961i − 0.139454i −0.997566 0.0697271i 0.977787π-0.977787\pi
0.997566 0.0697271i 0.0222129π-0.0222129\pi
88 −0.595174 −0.210426
99 −3.36786 −1.12262
1010 − 2.07061i − 0.654785i
1111 2.49810i 0.753206i 0.926375 + 0.376603i 0.122908π0.122908\pi
−0.926375 + 0.376603i 0.877092π0.877092\pi
1212 5.77226i 1.66631i
1313 4.68778 1.30016 0.650078 0.759868i 0.274736π-0.274736\pi
0.650078 + 0.759868i 0.274736π0.274736\pi
1414 0.763976i 0.204181i
1515 −2.52346 −0.651555
1616 −3.34250 −0.835626
1717 0 0
1818 6.97354 1.64368
1919 7.16938 1.64477 0.822384 0.568933i 0.192644π-0.192644\pi
0.822384 + 0.568933i 0.192644π0.192644\pi
2020 2.28744i 0.511487i
2121 0.931060 0.203174
2222 − 5.17260i − 1.10280i
2323 − 6.70055i − 1.39716i −0.715531 0.698581i 0.753815π-0.753815\pi
0.715531 0.698581i 0.246185π-0.246185\pi
2424 − 1.50190i − 0.306574i
2525 −1.00000 −0.200000
2626 −9.70657 −1.90361
2727 − 0.928289i − 0.178649i
2828 − 0.843976i − 0.159496i
2929 6.78165i 1.25932i 0.776870 + 0.629661i 0.216806π0.216806\pi
−0.776870 + 0.629661i 0.783194π0.783194\pi
3030 5.22511 0.953971
3131 4.81801i 0.865341i 0.901552 + 0.432670i 0.142429π0.142429\pi
−0.901552 + 0.432670i 0.857571π0.857571\pi
3232 8.11138 1.43390
3333 −6.30387 −1.09736
3434 0 0
3535 0.368961 0.0623658
3636 −7.70378 −1.28396
3737 − 1.93935i − 0.318827i −0.987212 0.159413i 0.949040π-0.949040\pi
0.987212 0.159413i 0.0509603π-0.0509603\pi
3838 −14.8450 −2.40818
3939 11.8294i 1.89422i
4040 − 0.595174i − 0.0941052i
4141 2.35972i 0.368527i 0.982877 + 0.184263i 0.0589900π0.0589900\pi
−0.982877 + 0.184263i 0.941010π0.941010\pi
4242 −1.92786 −0.297476
4343 11.7105 1.78584 0.892918 0.450218i 0.148654π-0.148654\pi
0.892918 + 0.450218i 0.148654π0.148654\pi
4444 5.71425i 0.861456i
4545 − 3.36786i − 0.502051i
4646 13.8743i 2.04565i
4747 1.65317 0.241140 0.120570 0.992705i 0.461528π-0.461528\pi
0.120570 + 0.992705i 0.461528π0.461528\pi
4848 − 8.43468i − 1.21744i
4949 6.86387 0.980553
5050 2.07061 0.292829
5151 0 0
5252 10.7230 1.48701
5353 6.81536 0.936162 0.468081 0.883686i 0.344946π-0.344946\pi
0.468081 + 0.883686i 0.344946π0.344946\pi
5454 1.92213i 0.261568i
5555 −2.49810 −0.336844
5656 0.219596i 0.0293447i
5757 18.0917i 2.39630i
5858 − 14.0422i − 1.84383i
5959 0.484372 0.0630598 0.0315299 0.999503i 0.489962π-0.489962\pi
0.0315299 + 0.999503i 0.489962π0.489962\pi
6060 −5.77226 −0.745196
6161 6.88307i 0.881286i 0.897682 + 0.440643i 0.145250π0.145250\pi
−0.897682 + 0.440643i 0.854750π0.854750\pi
6262 − 9.97624i − 1.26698i
6363 1.24261i 0.156554i
6464 −10.1105 −1.26381
6565 4.68778i 0.581447i
6666 13.0529 1.60670
6767 −1.87478 −0.229041 −0.114520 0.993421i 0.536533π-0.536533\pi
−0.114520 + 0.993421i 0.536533π0.536533\pi
6868 0 0
6969 16.9086 2.03556
7070 −0.763976 −0.0913126
7171 1.71958i 0.204077i 0.994780 + 0.102038i 0.0325365π0.0325365\pi
−0.994780 + 0.102038i 0.967464π0.967464\pi
7272 2.00446 0.236228
7373 − 0.286310i − 0.0335101i −0.999860 0.0167550i 0.994666π-0.994666\pi
0.999860 0.0167550i 0.00533355π-0.00533355\pi
7474 4.01564i 0.466809i
7575 − 2.52346i − 0.291384i
7676 16.3995 1.88115
7777 0.921703 0.105038
7878 − 24.4942i − 2.77342i
7979 − 5.38563i − 0.605930i −0.953002 0.302965i 0.902023π-0.902023\pi
0.953002 0.302965i 0.0979766π-0.0979766\pi
8080 − 3.34250i − 0.373703i
8181 −7.76109 −0.862343
8282 − 4.88608i − 0.539577i
8383 −9.94985 −1.09214 −0.546069 0.837740i 0.683876π-0.683876\pi
−0.546069 + 0.837740i 0.683876π0.683876\pi
8484 2.12974 0.232374
8585 0 0
8686 −24.2479 −2.61472
8787 −17.1132 −1.83473
8888 − 1.48680i − 0.158494i
8989 4.30781 0.456627 0.228314 0.973588i 0.426679π-0.426679\pi
0.228314 + 0.973588i 0.426679π0.426679\pi
9090 6.97354i 0.735076i
9191 − 1.72961i − 0.181312i
9292 − 15.3271i − 1.59796i
9393 −12.1581 −1.26073
9494 −3.42308 −0.353064
9595 7.16938i 0.735563i
9696 20.4688i 2.08908i
9797 − 12.7460i − 1.29416i −0.762424 0.647078i 0.775991π-0.775991\pi
0.762424 0.647078i 0.224009π-0.224009\pi
9898 −14.2124 −1.43567
9999 − 8.41327i − 0.845565i
100100 −2.28744 −0.228744
101101 −9.08552 −0.904043 −0.452021 0.892007i 0.649297π-0.649297\pi
−0.452021 + 0.892007i 0.649297π0.649297\pi
102102 0 0
103103 4.77811 0.470801 0.235401 0.971898i 0.424360π-0.424360\pi
0.235401 + 0.971898i 0.424360π0.424360\pi
104104 −2.79004 −0.273586
105105 0.931060i 0.0908621i
106106 −14.1120 −1.37068
107107 − 2.85209i − 0.275722i −0.990452 0.137861i 0.955977π-0.955977\pi
0.990452 0.137861i 0.0440227π-0.0440227\pi
108108 − 2.12340i − 0.204325i
109109 − 9.44863i − 0.905014i −0.891761 0.452507i 0.850530π-0.850530\pi
0.891761 0.452507i 0.149470π-0.149470\pi
110110 5.17260 0.493188
111111 4.89387 0.464506
112112 1.23325i 0.116532i
113113 10.6789i 1.00459i 0.864698 + 0.502293i 0.167510π0.167510\pi
−0.864698 + 0.502293i 0.832490π0.832490\pi
114114 − 37.4608i − 3.50853i
115115 6.70055 0.624830
116116 15.5126i 1.44031i
117117 −15.7878 −1.45958
118118 −1.00295 −0.0923287
119119 0 0
120120 1.50190 0.137104
121121 4.75949 0.432681
122122 − 14.2522i − 1.29033i
123123 −5.95468 −0.536915
124124 11.0209i 0.989707i
125125 − 1.00000i − 0.0894427i
126126 − 2.57297i − 0.229218i
127127 −15.7315 −1.39595 −0.697974 0.716124i 0.745915π-0.745915\pi
−0.697974 + 0.716124i 0.745915π0.745915\pi
128128 4.71221 0.416505
129129 29.5510i 2.60182i
130130 − 9.70657i − 0.851322i
131131 4.09160i 0.357485i 0.983896 + 0.178742i 0.0572029π0.0572029\pi
−0.983896 + 0.178742i 0.942797π0.942797\pi
132132 −14.4197 −1.25507
133133 − 2.64522i − 0.229370i
134134 3.88194 0.335348
135135 0.928289 0.0798944
136136 0 0
137137 −16.2144 −1.38529 −0.692644 0.721280i 0.743554π-0.743554\pi
−0.692644 + 0.721280i 0.743554π0.743554\pi
138138 −35.0112 −2.98035
139139 10.7071i 0.908168i 0.890959 + 0.454084i 0.150033π0.150033\pi
−0.890959 + 0.454084i 0.849967π0.849967\pi
140140 0.843976 0.0713290
141141 4.17172i 0.351322i
142142 − 3.56059i − 0.298798i
143143 11.7105i 0.979285i
144144 11.2571 0.938091
145145 −6.78165 −0.563186
146146 0.592838i 0.0490636i
147147 17.3207i 1.42859i
148148 − 4.43614i − 0.364649i
149149 11.1290 0.911719 0.455860 0.890052i 0.349332π-0.349332\pi
0.455860 + 0.890052i 0.349332π0.349332\pi
150150 5.22511i 0.426629i
151151 −16.0861 −1.30907 −0.654535 0.756032i 0.727135π-0.727135\pi
−0.654535 + 0.756032i 0.727135π0.727135\pi
152152 −4.26702 −0.346101
153153 0 0
154154 −1.90849 −0.153790
155155 −4.81801 −0.386992
156156 27.0591i 2.16646i
157157 2.63326 0.210157 0.105079 0.994464i 0.466491π-0.466491\pi
0.105079 + 0.994464i 0.466491π0.466491\pi
158158 11.1516i 0.887170i
159159 17.1983i 1.36392i
160160 8.11138i 0.641261i
161161 −2.47224 −0.194840
162162 16.0702 1.26259
163163 − 19.0937i − 1.49554i −0.663959 0.747769i 0.731125π-0.731125\pi
0.663959 0.747769i 0.268875π-0.268875\pi
164164 5.39772i 0.421491i
165165 − 6.30387i − 0.490755i
166166 20.6023 1.59905
167167 3.98742i 0.308556i 0.988028 + 0.154278i 0.0493051π0.0493051\pi
−0.988028 + 0.154278i 0.950695π0.950695\pi
168168 −0.554142 −0.0427530
169169 8.97524 0.690403
170170 0 0
171171 −24.1455 −1.84645
172172 26.7871 2.04250
173173 − 14.6078i − 1.11061i −0.831646 0.555307i 0.812601π-0.812601\pi
0.831646 0.555307i 0.187399π-0.187399\pi
174174 35.4349 2.68631
175175 0.368961i 0.0278908i
176176 − 8.34991i − 0.629398i
177177 1.22229i 0.0918733i
178178 −8.91981 −0.668568
179179 −14.7452 −1.10211 −0.551054 0.834469i 0.685774π-0.685774\pi
−0.551054 + 0.834469i 0.685774π0.685774\pi
180180 − 7.70378i − 0.574206i
181181 − 11.2615i − 0.837059i −0.908203 0.418529i 0.862546π-0.862546\pi
0.908203 0.418529i 0.137454π-0.137454\pi
182182 3.58135i 0.265467i
183183 −17.3692 −1.28397
184184 3.98799i 0.293999i
185185 1.93935 0.142584
186186 25.1747 1.84590
187187 0 0
188188 3.78153 0.275797
189189 −0.342503 −0.0249134
190190 − 14.8450i − 1.07697i
191191 −21.4934 −1.55521 −0.777604 0.628754i 0.783565π-0.783565\pi
−0.777604 + 0.628754i 0.783565π0.783565\pi
192192 − 25.5135i − 1.84128i
193193 1.42208i 0.102363i 0.998689 + 0.0511817i 0.0162988π0.0162988\pi
−0.998689 + 0.0511817i 0.983701π0.983701\pi
194194 26.3919i 1.89483i
195195 −11.8294 −0.847123
196196 15.7007 1.12148
197197 17.4127i 1.24060i 0.784363 + 0.620302i 0.212990π0.212990\pi
−0.784363 + 0.620302i 0.787010π0.787010\pi
198198 17.4206i 1.23803i
199199 17.8876i 1.26802i 0.773325 + 0.634010i 0.218592π0.218592\pi
−0.773325 + 0.634010i 0.781408π0.781408\pi
200200 0.595174 0.0420851
201201 − 4.73093i − 0.333694i
202202 18.8126 1.32365
203203 2.50217 0.175618
204204 0 0
205205 −2.35972 −0.164810
206206 −9.89362 −0.689321
207207 22.5665i 1.56848i
208208 −15.6689 −1.08644
209209 17.9098i 1.23885i
210210 − 1.92786i − 0.133035i
211211 − 7.17979i − 0.494277i −0.968980 0.247139i 0.920510π-0.920510\pi
0.968980 0.247139i 0.0794903π-0.0794903\pi
212212 15.5897 1.07071
213213 −4.33930 −0.297324
214214 5.90557i 0.403697i
215215 11.7105i 0.798651i
216216 0.552493i 0.0375924i
217217 1.77766 0.120675
218218 19.5645i 1.32507i
219219 0.722494 0.0488216
220220 −5.71425 −0.385255
221221 0 0
222222 −10.1333 −0.680104
223223 3.93334 0.263396 0.131698 0.991290i 0.457957π-0.457957\pi
0.131698 + 0.991290i 0.457957π0.457957\pi
224224 − 2.99278i − 0.199964i
225225 3.36786 0.224524
226226 − 22.1119i − 1.47086i
227227 − 3.47984i − 0.230965i −0.993310 0.115483i 0.963159π-0.963159\pi
0.993310 0.115483i 0.0368414π-0.0368414\pi
228228 41.3835i 2.74069i
229229 −4.41164 −0.291529 −0.145765 0.989319i 0.546564π-0.546564\pi
−0.145765 + 0.989319i 0.546564π0.546564\pi
230230 −13.8743 −0.914841
231231 2.32588i 0.153032i
232232 − 4.03626i − 0.264993i
233233 15.6666i 1.02635i 0.858284 + 0.513175i 0.171531π0.171531\pi
−0.858284 + 0.513175i 0.828469π0.828469\pi
234234 32.6904 2.13704
235235 1.65317i 0.107841i
236236 1.10797 0.0721227
237237 13.5904 0.882793
238238 0 0
239239 −6.59116 −0.426346 −0.213173 0.977014i 0.568380π-0.568380\pi
−0.213173 + 0.977014i 0.568380π0.568380\pi
240240 8.43468 0.544456
241241 4.38042i 0.282167i 0.989998 + 0.141084i 0.0450587π0.0450587\pi
−0.989998 + 0.141084i 0.954941π0.954941\pi
242242 −9.85506 −0.633507
243243 − 22.3697i − 1.43502i
244244 15.7446i 1.00794i
245245 6.86387i 0.438516i
246246 12.3298 0.786121
247247 33.6084 2.13845
248248 − 2.86755i − 0.182090i
249249 − 25.1081i − 1.59116i
250250 2.07061i 0.130957i
251251 4.30290 0.271597 0.135798 0.990736i 0.456640π-0.456640\pi
0.135798 + 0.990736i 0.456640π0.456640\pi
252252 2.84240i 0.179054i
253253 16.7387 1.05235
254254 32.5739 2.04387
255255 0 0
256256 10.4639 0.653991
257257 −10.5366 −0.657254 −0.328627 0.944460i 0.606586π-0.606586\pi
−0.328627 + 0.944460i 0.606586π0.606586\pi
258258 − 61.1888i − 3.80945i
259259 −0.715544 −0.0444618
260260 10.7230i 0.665012i
261261 − 22.8397i − 1.41374i
262262 − 8.47213i − 0.523410i
263263 −13.6253 −0.840173 −0.420086 0.907484i 0.638000π-0.638000\pi
−0.420086 + 0.907484i 0.638000π0.638000\pi
264264 3.75189 0.230913
265265 6.81536i 0.418664i
266266 5.47723i 0.335831i
267267 10.8706i 0.665270i
268268 −4.28844 −0.261958
269269 15.6073i 0.951593i 0.879555 + 0.475796i 0.157840π0.157840\pi
−0.879555 + 0.475796i 0.842160π0.842160\pi
270270 −1.92213 −0.116977
271271 −9.66560 −0.587143 −0.293572 0.955937i 0.594844π-0.594844\pi
−0.293572 + 0.955937i 0.594844π0.594844\pi
272272 0 0
273273 4.36460 0.264158
274274 33.5737 2.02826
275275 − 2.49810i − 0.150641i
276276 38.6774 2.32810
277277 1.35066i 0.0811535i 0.999176 + 0.0405767i 0.0129195π0.0129195\pi
−0.999176 + 0.0405767i 0.987080π0.987080\pi
278278 − 22.1703i − 1.32969i
279279 − 16.2264i − 0.971450i
280280 −0.219596 −0.0131234
281281 15.8582 0.946020 0.473010 0.881057i 0.343167π-0.343167\pi
0.473010 + 0.881057i 0.343167π0.343167\pi
282282 − 8.63802i − 0.514387i
283283 − 29.3390i − 1.74402i −0.489488 0.872010i 0.662816π-0.662816\pi
0.489488 0.872010i 0.337184π-0.337184\pi
284284 3.93344i 0.233407i
285285 −18.0917 −1.07166
286286 − 24.2480i − 1.43381i
287287 0.870646 0.0513926
288288 −27.3180 −1.60973
289289 0 0
290290 14.0422 0.824585
291291 32.1639 1.88548
292292 − 0.654917i − 0.0383261i
293293 20.6478 1.20626 0.603129 0.797644i 0.293920π-0.293920\pi
0.603129 + 0.797644i 0.293920π0.293920\pi
294294 − 35.8645i − 2.09166i
295295 0.484372i 0.0282012i
296296 1.15425i 0.0670894i
297297 2.31896 0.134560
298298 −23.0437 −1.33489
299299 − 31.4107i − 1.81653i
300300 − 5.77226i − 0.333262i
301301 − 4.32073i − 0.249042i
302302 33.3081 1.91667
303303 − 22.9270i − 1.31712i
304304 −23.9637 −1.37441
305305 −6.88307 −0.394123
306306 0 0
307307 16.3437 0.932787 0.466393 0.884577i 0.345553π-0.345553\pi
0.466393 + 0.884577i 0.345553π0.345553\pi
308308 2.10834 0.120134
309309 12.0574i 0.685921i
310310 9.97624 0.566613
311311 − 35.1101i − 1.99091i −0.0952411 0.995454i 0.530362π-0.530362\pi
0.0952411 0.995454i 0.469638π-0.469638\pi
312312 − 7.04056i − 0.398593i
313313 31.3134i 1.76994i 0.465651 + 0.884968i 0.345820π0.345820\pi
−0.465651 + 0.884968i 0.654180π0.654180\pi
314314 −5.45247 −0.307701
315315 −1.24261 −0.0700132
316316 − 12.3193i − 0.693014i
317317 − 12.5497i − 0.704862i −0.935838 0.352431i 0.885355π-0.885355\pi
0.935838 0.352431i 0.114645π-0.114645\pi
318318 − 35.6111i − 1.99697i
319319 −16.9413 −0.948528
320320 − 10.1105i − 0.565195i
321321 7.19714 0.401705
322322 5.11906 0.285274
323323 0 0
324324 −17.7530 −0.986278
325325 −4.68778 −0.260031
326326 39.5358i 2.18968i
327327 23.8433 1.31854
328328 − 1.40445i − 0.0775475i
329329 − 0.609957i − 0.0336280i
330330 13.0529i 0.718537i
331331 0.497450 0.0273423 0.0136712 0.999907i 0.495648π-0.495648\pi
0.0136712 + 0.999907i 0.495648π0.495648\pi
332332 −22.7597 −1.24910
333333 6.53146i 0.357922i
334334 − 8.25640i − 0.451770i
335335 − 1.87478i − 0.102430i
336336 −3.11207 −0.169777
337337 − 33.5675i − 1.82854i −0.405108 0.914269i 0.632766π-0.632766\pi
0.405108 0.914269i 0.367234π-0.367234\pi
338338 −18.5843 −1.01085
339339 −26.9478 −1.46360
340340 0 0
341341 −12.0359 −0.651780
342342 49.9959 2.70347
343343 − 5.11523i − 0.276196i
344344 −6.96979 −0.375786
345345 16.9086i 0.910328i
346346 30.2472i 1.62610i
347347 26.3010i 1.41191i 0.708257 + 0.705955i 0.249482π0.249482\pi
−0.708257 + 0.705955i 0.750518π0.750518\pi
348348 −39.1455 −2.09842
349349 −16.7865 −0.898560 −0.449280 0.893391i 0.648319π-0.648319\pi
−0.449280 + 0.893391i 0.648319π0.648319\pi
350350 − 0.763976i − 0.0408362i
351351 − 4.35161i − 0.232272i
352352 20.2630i 1.08002i
353353 23.4532 1.24829 0.624144 0.781309i 0.285448π-0.285448\pi
0.624144 + 0.781309i 0.285448π0.285448\pi
354354 − 2.53090i − 0.134516i
355355 −1.71958 −0.0912660
356356 9.85386 0.522253
357357 0 0
358358 30.5316 1.61365
359359 −18.1568 −0.958279 −0.479139 0.877739i 0.659051π-0.659051\pi
−0.479139 + 0.877739i 0.659051π0.659051\pi
360360 2.00446i 0.105644i
361361 32.4000 1.70526
362362 23.3182i 1.22558i
363363 12.0104i 0.630382i
364364 − 3.95637i − 0.207370i
365365 0.286310 0.0149862
366366 35.9648 1.87991
367367 − 13.6904i − 0.714635i −0.933983 0.357318i 0.883691π-0.883691\pi
0.933983 0.357318i 0.116309π-0.116309\pi
368368 22.3966i 1.16750i
369369 − 7.94723i − 0.413716i
370370 −4.01564 −0.208763
371371 − 2.51460i − 0.130552i
372372 −27.8109 −1.44193
373373 −10.1594 −0.526035 −0.263018 0.964791i 0.584718π-0.584718\pi
−0.263018 + 0.964791i 0.584718π0.584718\pi
374374 0 0
375375 2.52346 0.130311
376376 −0.983925 −0.0507421
377377 31.7909i 1.63731i
378378 0.709190 0.0364768
379379 4.16411i 0.213896i 0.994265 + 0.106948i 0.0341078π0.0341078\pi
−0.994265 + 0.106948i 0.965892π0.965892\pi
380380 16.3995i 0.841277i
381381 − 39.6979i − 2.03379i
382382 44.5045 2.27705
383383 −0.813273 −0.0415563 −0.0207782 0.999784i 0.506614π-0.506614\pi
−0.0207782 + 0.999784i 0.506614π0.506614\pi
384384 11.8911i 0.606815i
385385 0.921703i 0.0469743i
386386 − 2.94457i − 0.149875i
387387 −39.4394 −2.00482
388388 − 29.1556i − 1.48015i
389389 −3.11676 −0.158026 −0.0790131 0.996874i 0.525177π-0.525177\pi
−0.0790131 + 0.996874i 0.525177π0.525177\pi
390390 24.4942 1.24031
391391 0 0
392392 −4.08519 −0.206333
393393 −10.3250 −0.520828
394394 − 36.0550i − 1.81642i
395395 5.38563 0.270980
396396 − 19.2448i − 0.967089i
397397 − 16.4046i − 0.823322i −0.911337 0.411661i 0.864949π-0.864949\pi
0.911337 0.411661i 0.135051π-0.135051\pi
398398 − 37.0384i − 1.85657i
399399 6.67512 0.334174
400400 3.34250 0.167125
401401 14.4451i 0.721356i 0.932690 + 0.360678i 0.117455π0.117455\pi
−0.932690 + 0.360678i 0.882545π0.882545\pi
402402 9.79593i 0.488576i
403403 22.5858i 1.12508i
404404 −20.7826 −1.03397
405405 − 7.76109i − 0.385652i
406406 −5.18102 −0.257130
407407 4.84469 0.240142
408408 0 0
409409 32.2867 1.59648 0.798238 0.602342i 0.205765π-0.205765\pi
0.798238 + 0.602342i 0.205765π0.205765\pi
410410 4.88608 0.241306
411411 − 40.9164i − 2.01826i
412412 10.9296 0.538464
413413 − 0.178714i − 0.00879396i
414414 − 46.7266i − 2.29649i
415415 − 9.94985i − 0.488419i
416416 38.0243 1.86430
417417 −27.0191 −1.32313
418418 − 37.0843i − 1.81385i
419419 − 18.8776i − 0.922230i −0.887340 0.461115i 0.847450π-0.847450\pi
0.887340 0.461115i 0.152550π-0.152550\pi
420420 2.12974i 0.103921i
421421 2.06006 0.100401 0.0502006 0.998739i 0.484014π-0.484014\pi
0.0502006 + 0.998739i 0.484014π0.484014\pi
422422 14.8666i 0.723693i
423423 −5.56766 −0.270709
424424 −4.05632 −0.196993
425425 0 0
426426 8.98502 0.435326
427427 2.53958 0.122899
428428 − 6.52398i − 0.315348i
429429 −29.5511 −1.42674
430430 − 24.2479i − 1.16934i
431431 3.72759i 0.179552i 0.995962 + 0.0897759i 0.0286151π0.0286151\pi
−0.995962 + 0.0897759i 0.971385π0.971385\pi
432432 3.10281i 0.149284i
433433 27.9634 1.34383 0.671917 0.740626i 0.265471π-0.265471\pi
0.671917 + 0.740626i 0.265471π0.265471\pi
434434 −3.68085 −0.176686
435435 − 17.1132i − 0.820517i
436436 − 21.6132i − 1.03508i
437437 − 48.0388i − 2.29801i
438438 −1.49600 −0.0714819
439439 12.5444i 0.598711i 0.954142 + 0.299356i 0.0967717π0.0967717\pi
−0.954142 + 0.299356i 0.903228π0.903228\pi
440440 1.48680 0.0708806
441441 −23.1166 −1.10079
442442 0 0
443443 −19.9529 −0.947991 −0.473995 0.880527i 0.657189π-0.657189\pi
−0.473995 + 0.880527i 0.657189π0.657189\pi
444444 11.1944 0.531264
445445 4.30781i 0.204210i
446446 −8.14442 −0.385649
447447 28.0835i 1.32830i
448448 3.73039i 0.176244i
449449 20.1861i 0.952640i 0.879272 + 0.476320i 0.158030π0.158030\pi
−0.879272 + 0.476320i 0.841970π0.841970\pi
450450 −6.97354 −0.328736
451451 −5.89483 −0.277577
452452 24.4273i 1.14896i
453453 − 40.5927i − 1.90721i
454454 7.20540i 0.338166i
455455 1.72961 0.0810852
456456 − 10.7677i − 0.504243i
457457 37.7391 1.76536 0.882681 0.469973i 0.155737π-0.155737\pi
0.882681 + 0.469973i 0.155737π0.155737\pi
458458 9.13479 0.426841
459459 0 0
460460 15.3271 0.714630
461461 38.1740 1.77794 0.888970 0.457966i 0.151422π-0.151422\pi
0.888970 + 0.457966i 0.151422π0.151422\pi
462462 − 4.81600i − 0.224061i
463463 −13.1481 −0.611044 −0.305522 0.952185i 0.598831π-0.598831\pi
−0.305522 + 0.952185i 0.598831π0.598831\pi
464464 − 22.6677i − 1.05232i
465465 − 12.1581i − 0.563817i
466466 − 32.4394i − 1.50272i
467467 16.2167 0.750422 0.375211 0.926940i 0.377570π-0.377570\pi
0.375211 + 0.926940i 0.377570π0.377570\pi
468468 −36.1136 −1.66935
469469 0.691720i 0.0319407i
470470 − 3.42308i − 0.157895i
471471 6.64494i 0.306183i
472472 −0.288285 −0.0132694
473473 29.2541i 1.34510i
474474 −28.1405 −1.29254
475475 −7.16938 −0.328954
476476 0 0
477477 −22.9532 −1.05096
478478 13.6477 0.624233
479479 2.07499i 0.0948086i 0.998876 + 0.0474043i 0.0150949π0.0150949\pi
−0.998876 + 0.0474043i 0.984905π0.984905\pi
480480 −20.4688 −0.934267
481481 − 9.09123i − 0.414525i
482482 − 9.07015i − 0.413134i
483483 − 6.23862i − 0.283867i
484484 10.8870 0.494865
485485 12.7460 0.578764
486486 46.3189i 2.10107i
487487 24.0260i 1.08872i 0.838851 + 0.544361i 0.183228π0.183228\pi
−0.838851 + 0.544361i 0.816772π0.816772\pi
488488 − 4.09662i − 0.185445i
489489 48.1824 2.17888
490490 − 14.2124i − 0.642051i
491491 22.3803 1.01001 0.505005 0.863116i 0.331491π-0.331491\pi
0.505005 + 0.863116i 0.331491π0.331491\pi
492492 −13.6210 −0.614080
493493 0 0
494494 −69.5901 −3.13100
495495 8.41327 0.378148
496496 − 16.1042i − 0.723101i
497497 0.634459 0.0284594
498498 51.9891i 2.32969i
499499 3.69807i 0.165548i 0.996568 + 0.0827742i 0.0263780π0.0263780\pi
−0.996568 + 0.0827742i 0.973622π0.973622\pi
500500 − 2.28744i − 0.102297i
501501 −10.0621 −0.449541
502502 −8.90965 −0.397657
503503 − 18.4617i − 0.823166i −0.911372 0.411583i 0.864976π-0.864976\pi
0.911372 0.411583i 0.135024π-0.135024\pi
504504 − 0.739569i − 0.0329430i
505505 − 9.08552i − 0.404300i
506506 −34.6593 −1.54079
507507 22.6487i 1.00586i
508508 −35.9849 −1.59657
509509 −17.3588 −0.769415 −0.384708 0.923039i 0.625698π-0.625698\pi
−0.384708 + 0.923039i 0.625698π0.625698\pi
510510 0 0
511511 −0.105637 −0.00467312
512512 −31.0910 −1.37404
513513 − 6.65525i − 0.293837i
514514 21.8172 0.962314
515515 4.77811i 0.210549i
516516 67.5962i 2.97576i
517517 4.12980i 0.181628i
518518 1.48162 0.0650984
519519 36.8623 1.61808
520520 − 2.79004i − 0.122351i
521521 − 13.9650i − 0.611819i −0.952061 0.305909i 0.901040π-0.901040\pi
0.952061 0.305909i 0.0989604π-0.0989604\pi
522522 47.2921i 2.06992i
523523 −5.59638 −0.244712 −0.122356 0.992486i 0.539045π-0.539045\pi
−0.122356 + 0.992486i 0.539045π0.539045\pi
524524 9.35929i 0.408862i
525525 −0.931060 −0.0406348
526526 28.2127 1.23013
527527 0 0
528528 21.0707 0.916984
529529 −21.8974 −0.952062
530530 − 14.1120i − 0.612985i
531531 −1.63130 −0.0707923
532532 − 6.05078i − 0.262335i
533533 11.0619i 0.479142i
534534 − 22.5088i − 0.974052i
535535 2.85209 0.123307
536536 1.11582 0.0481960
537537 − 37.2090i − 1.60569i
538538 − 32.3166i − 1.39327i
539539 17.1466i 0.738558i
540540 2.12340 0.0913768
541541 39.0998i 1.68103i 0.541787 + 0.840516i 0.317748π0.317748\pi
−0.541787 + 0.840516i 0.682252π0.682252\pi
542542 20.0137 0.859662
543543 28.4179 1.21953
544544 0 0
545545 9.44863 0.404735
546546 −9.03740 −0.386765
547547 29.3984i 1.25698i 0.777816 + 0.628492i 0.216328π0.216328\pi
−0.777816 + 0.628492i 0.783672π0.783672\pi
548548 −37.0894 −1.58438
549549 − 23.1812i − 0.989351i
550550 5.17260i 0.220561i
551551 48.6202i 2.07129i
552552 −10.0635 −0.428333
553553 −1.98709 −0.0844995
554554 − 2.79670i − 0.118820i
555555 4.89387i 0.207733i
556556 24.4919i 1.03869i
557557 27.9399 1.18385 0.591927 0.805992i 0.298368π-0.298368\pi
0.591927 + 0.805992i 0.298368π0.298368\pi
558558 33.5986i 1.42234i
559559 54.8963 2.32186
560560 −1.23325 −0.0521145
561561 0 0
562562 −32.8362 −1.38511
563563 11.2994 0.476213 0.238106 0.971239i 0.423473π-0.423473\pi
0.238106 + 0.971239i 0.423473π0.423473\pi
564564 9.54256i 0.401814i
565565 −10.6789 −0.449264
566566 60.7496i 2.55350i
567567 2.86354i 0.120257i
568568 − 1.02345i − 0.0429430i
569569 12.1204 0.508114 0.254057 0.967189i 0.418235π-0.418235\pi
0.254057 + 0.967189i 0.418235π0.418235\pi
570570 37.4608 1.56906
571571 − 10.9497i − 0.458232i −0.973399 0.229116i 0.926416π-0.926416\pi
0.973399 0.229116i 0.0735836π-0.0735836\pi
572572 26.7871i 1.12003i
573573 − 54.2378i − 2.26582i
574574 −1.80277 −0.0752462
575575 6.70055i 0.279432i
576576 34.0508 1.41878
577577 −9.09883 −0.378789 −0.189395 0.981901i 0.560653π-0.560653\pi
−0.189395 + 0.981901i 0.560653π0.560653\pi
578578 0 0
579579 −3.58856 −0.149135
580580 −15.5126 −0.644126
581581 3.67111i 0.152303i
582582 −66.5991 −2.76062
583583 17.0255i 0.705123i
584584 0.170404i 0.00705138i
585585 − 15.7878i − 0.652745i
586586 −42.7536 −1.76614
587587 −11.2991 −0.466362 −0.233181 0.972433i 0.574913π-0.574913\pi
−0.233181 + 0.972433i 0.574913π0.574913\pi
588588 39.6201i 1.63390i
589589 34.5422i 1.42329i
590590 − 1.00295i − 0.0412907i
591591 −43.9403 −1.80746
592592 6.48228i 0.266420i
593593 −42.8620 −1.76013 −0.880066 0.474851i 0.842502π-0.842502\pi
−0.880066 + 0.474851i 0.842502π0.842502\pi
594594 −4.80167 −0.197015
595595 0 0
596596 25.4568 1.04275
597597 −45.1388 −1.84741
598598 65.0394i 2.65966i
599599 −21.0108 −0.858478 −0.429239 0.903191i 0.641218π-0.641218\pi
−0.429239 + 0.903191i 0.641218π0.641218\pi
600600 1.50190i 0.0613147i
601601 19.4501i 0.793387i 0.917951 + 0.396693i 0.129842π0.129842\pi
−0.917951 + 0.396693i 0.870158π0.870158\pi
602602 8.94655i 0.364634i
603603 6.31399 0.257126
604604 −36.7960 −1.49721
605605 4.75949i 0.193501i
606606 47.4729i 1.92845i
607607 27.4308i 1.11338i 0.830719 + 0.556691i 0.187929π0.187929\pi
−0.830719 + 0.556691i 0.812071π0.812071\pi
608608 58.1535 2.35844
609609 6.31412i 0.255861i
610610 14.2522 0.577053
611611 7.74971 0.313520
612612 0 0
613613 0.297602 0.0120200 0.00601001 0.999982i 0.498087π-0.498087\pi
0.00601001 + 0.999982i 0.498087π0.498087\pi
614614 −33.8416 −1.36573
615615 − 5.95468i − 0.240116i
616616 −0.548573 −0.0221026
617617 − 12.6680i − 0.509995i −0.966942 0.254997i 0.917925π-0.917925\pi
0.966942 0.254997i 0.0820746π-0.0820746\pi
618618 − 24.9662i − 1.00429i
619619 9.75795i 0.392205i 0.980583 + 0.196103i 0.0628285π0.0628285\pi
−0.980583 + 0.196103i 0.937171π0.937171\pi
620620 −11.0209 −0.442610
621621 −6.22005 −0.249602
622622 72.6993i 2.91498i
623623 − 1.58942i − 0.0636786i
624624 − 39.5399i − 1.58286i
625625 1.00000 0.0400000
626626 − 64.8379i − 2.59144i
627627 −45.1948 −1.80491
628628 6.02343 0.240361
629629 0 0
630630 2.57297 0.102509
631631 −40.9574 −1.63049 −0.815244 0.579117i 0.803397π-0.803397\pi
−0.815244 + 0.579117i 0.803397π0.803397\pi
632632 3.20538i 0.127503i
633633 18.1179 0.720123
634634 25.9856i 1.03202i
635635 − 15.7315i − 0.624287i
636636 39.3401i 1.55994i
637637 32.1763 1.27487
638638 35.0788 1.38878
639639 − 5.79132i − 0.229101i
640640 4.71221i 0.186267i
641641 − 3.96498i − 0.156607i −0.996930 0.0783037i 0.975050π-0.975050\pi
0.996930 0.0783037i 0.0249504π-0.0249504\pi
642642 −14.9025 −0.588154
643643 15.4283i 0.608433i 0.952603 + 0.304217i 0.0983947π0.0983947\pi
−0.952603 + 0.304217i 0.901605π0.901605\pi
644644 −5.65511 −0.222842
645645 −29.5510 −1.16357
646646 0 0
647647 −10.5735 −0.415687 −0.207843 0.978162i 0.566644π-0.566644\pi
−0.207843 + 0.978162i 0.566644π0.566644\pi
648648 4.61919 0.181459
649649 1.21001i 0.0474971i
650650 9.70657 0.380723
651651 4.48586i 0.175815i
652652 − 43.6758i − 1.71048i
653653 − 14.5705i − 0.570189i −0.958499 0.285094i 0.907975π-0.907975\pi
0.958499 0.285094i 0.0920250π-0.0920250\pi
654654 −49.3702 −1.93053
655655 −4.09160 −0.159872
656656 − 7.88738i − 0.307951i
657657 0.964254i 0.0376191i
658658 1.26298i 0.0492363i
659659 −23.7883 −0.926660 −0.463330 0.886186i 0.653346π-0.653346\pi
−0.463330 + 0.886186i 0.653346π0.653346\pi
660660 − 14.4197i − 0.561286i
661661 −9.91502 −0.385650 −0.192825 0.981233i 0.561765π-0.561765\pi
−0.192825 + 0.981233i 0.561765π0.561765\pi
662662 −1.03003 −0.0400331
663663 0 0
664664 5.92189 0.229814
665665 2.64522 0.102577
666666 − 13.5241i − 0.524049i
667667 45.4408 1.75948
668668 9.12097i 0.352901i
669669 9.92562i 0.383747i
670670 3.88194i 0.149972i
671671 −17.1946 −0.663790
672672 7.55218 0.291331
673673 − 3.97532i − 0.153237i −0.997060 0.0766186i 0.975588π-0.975588\pi
0.997060 0.0766186i 0.0244124π-0.0244124\pi
674674 69.5053i 2.67724i
675675 0.928289i 0.0357299i
676676 20.5303 0.789627
677677 − 20.3678i − 0.782800i −0.920221 0.391400i 0.871991π-0.871991\pi
0.920221 0.391400i 0.128009π-0.128009\pi
678678 55.7984 2.14293
679679 −4.70276 −0.180475
680680 0 0
681681 8.78125 0.336498
682682 24.9217 0.954300
683683 28.9372i 1.10725i 0.832766 + 0.553625i 0.186756π0.186756\pi
−0.832766 + 0.553625i 0.813244π0.813244\pi
684684 −55.2313 −2.11182
685685 − 16.2144i − 0.619520i
686686 10.5917i 0.404391i
687687 − 11.1326i − 0.424735i
688688 −39.1424 −1.49229
689689 31.9489 1.21716
690690 − 35.0112i − 1.33285i
691691 5.32679i 0.202641i 0.994854 + 0.101320i 0.0323067π0.0323067\pi
−0.994854 + 0.101320i 0.967693π0.967693\pi
692692 − 33.4145i − 1.27023i
693693 −3.10417 −0.117918
694694 − 54.4591i − 2.06724i
695695 −10.7071 −0.406145
696696 10.1854 0.386075
697697 0 0
698698 34.7583 1.31562
699699 −39.5340 −1.49531
700700 0.843976i 0.0318993i
701701 42.2699 1.59651 0.798256 0.602318i 0.205756π-0.205756\pi
0.798256 + 0.602318i 0.205756π0.205756\pi
702702 9.01050i 0.340079i
703703 − 13.9039i − 0.524396i
704704 − 25.2571i − 0.951913i
705705 −4.17172 −0.157116
706706 −48.5625 −1.82768
707707 3.35220i 0.126073i
708708 2.79592i 0.105077i
709709 48.3638i 1.81634i 0.418603 + 0.908169i 0.362520π0.362520\pi
−0.418603 + 0.908169i 0.637480π0.637480\pi
710710 3.56059 0.133627
711711 18.1381i 0.680230i
712712 −2.56390 −0.0960861
713713 32.2834 1.20902
714714 0 0
715715 −11.7105 −0.437949
716716 −33.7288 −1.26050
717717 − 16.6325i − 0.621153i
718718 37.5957 1.40306
719719 − 8.24579i − 0.307516i −0.988109 0.153758i 0.950862π-0.950862\pi
0.988109 0.153758i 0.0491376π-0.0491376\pi
720720 11.2571i 0.419527i
721721 − 1.76294i − 0.0656552i
722722 −67.0878 −2.49675
723723 −11.0538 −0.411096
724724 − 25.7599i − 0.957360i
725725 − 6.78165i − 0.251864i
726726 − 24.8689i − 0.922970i
727727 −6.46089 −0.239621 −0.119811 0.992797i 0.538229π-0.538229\pi
−0.119811 + 0.992797i 0.538229π0.538229\pi
728728 1.02942i 0.0381527i
729729 33.1658 1.22836
730730 −0.592838 −0.0219419
731731 0 0
732732 −39.7309 −1.46850
733733 15.9605 0.589515 0.294758 0.955572i 0.404761π-0.404761\pi
0.294758 + 0.955572i 0.404761π0.404761\pi
734734 28.3476i 1.04633i
735735 −17.3207 −0.638884
736736 − 54.3507i − 2.00339i
737737 − 4.68339i − 0.172515i
738738 16.4556i 0.605740i
739739 0.760943 0.0279918 0.0139959 0.999902i 0.495545π-0.495545\pi
0.0139959 + 0.999902i 0.495545π0.495545\pi
740740 4.43614 0.163076
741741 84.8096i 3.11556i
742742 5.20677i 0.191147i
743743 − 39.3288i − 1.44283i −0.692501 0.721417i 0.743491π-0.743491\pi
0.692501 0.721417i 0.256509π-0.256509\pi
744744 7.23617 0.265291
745745 11.1290i 0.407733i
746746 21.0362 0.770191
747747 33.5097 1.22606
748748 0 0
749749 −1.05231 −0.0384506
750750 −5.22511 −0.190794
751751 − 33.7907i − 1.23304i −0.787339 0.616521i 0.788542π-0.788542\pi
0.787339 0.616521i 0.211458π-0.211458\pi
752752 −5.52574 −0.201503
753753 10.8582i 0.395695i
754754 − 65.8266i − 2.39726i
755755 − 16.0861i − 0.585434i
756756 −0.783454 −0.0284939
757757 −23.0663 −0.838359 −0.419180 0.907903i 0.637682π-0.637682\pi
−0.419180 + 0.907903i 0.637682π0.637682\pi
758758 − 8.62225i − 0.313174i
759759 42.2394i 1.53319i
760760 − 4.26702i − 0.154781i
761761 −22.8435 −0.828076 −0.414038 0.910260i 0.635882π-0.635882\pi
−0.414038 + 0.910260i 0.635882π0.635882\pi
762762 82.1990i 2.97776i
763763 −3.48618 −0.126208
764764 −49.1648 −1.77872
765765 0 0
766766 1.68397 0.0608445
767767 2.27063 0.0819876
768768 26.4052i 0.952814i
769769 −3.68785 −0.132987 −0.0664936 0.997787i 0.521181π-0.521181\pi
−0.0664936 + 0.997787i 0.521181π0.521181\pi
770770 − 1.90849i − 0.0687772i
771771 − 26.5887i − 0.957567i
772772 3.25292i 0.117075i
773773 5.04027 0.181286 0.0906430 0.995883i 0.471108π-0.471108\pi
0.0906430 + 0.995883i 0.471108π0.471108\pi
774774 81.6638 2.93534
775775 − 4.81801i − 0.173068i
776776 7.58606i 0.272324i
777777 − 1.80565i − 0.0647773i
778778 6.45361 0.231373
779779 16.9178i 0.606141i
780780 −27.0591 −0.968871
781781 −4.29569 −0.153712
782782 0 0
783783 6.29533 0.224977
784784 −22.9425 −0.819375
785785 2.63326i 0.0939852i
786786 21.3791 0.762567
787787 13.6490i 0.486534i 0.969959 + 0.243267i 0.0782191π0.0782191\pi
−0.969959 + 0.243267i 0.921781π0.921781\pi
788788 39.8305i 1.41890i
789789 − 34.3830i − 1.22407i
790790 −11.1516 −0.396754
791791 3.94010 0.140094
792792 5.00735i 0.177929i
793793 32.2663i 1.14581i
794794 33.9675i 1.20546i
795795 −17.1983 −0.609961
796796 40.9169i 1.45026i
797797 16.6869 0.591081 0.295541 0.955330i 0.404500π-0.404500\pi
0.295541 + 0.955330i 0.404500π0.404500\pi
798798 −13.8216 −0.489279
799799 0 0
800800 −8.11138 −0.286780
801801 −14.5081 −0.512619
802802 − 29.9103i − 1.05617i
803803 0.715232 0.0252400
804804 − 10.8217i − 0.381652i
805805 − 2.47224i − 0.0871352i
806806 − 46.7664i − 1.64728i
807807 −39.3844 −1.38640
808808 5.40746 0.190234
809809 − 30.7399i − 1.08076i −0.841422 0.540379i 0.818281π-0.818281\pi
0.841422 0.540379i 0.181719π-0.181719\pi
810810 16.0702i 0.564650i
811811 − 21.3840i − 0.750894i −0.926844 0.375447i 0.877489π-0.877489\pi
0.926844 0.375447i 0.122511π-0.122511\pi
812812 5.72355 0.200857
813813 − 24.3908i − 0.855422i
814814 −10.0315 −0.351603
815815 19.0937 0.668825
816816 0 0
817817 83.9571 2.93729
818818 −66.8534 −2.33747
819819 5.82508i 0.203545i
820820 −5.39772 −0.188497
821821 11.9090i 0.415626i 0.978169 + 0.207813i 0.0666345π0.0666345\pi
−0.978169 + 0.207813i 0.933366π0.933366\pi
822822 84.7220i 2.95502i
823823 − 19.5943i − 0.683015i −0.939879 0.341507i 0.889063π-0.889063\pi
0.939879 0.341507i 0.110937π-0.110937\pi
824824 −2.84381 −0.0990686
825825 6.30387 0.219472
826826 0.370048i 0.0128756i
827827 − 35.0907i − 1.22022i −0.792316 0.610111i 0.791125π-0.791125\pi
0.792316 0.610111i 0.208875π-0.208875\pi
828828 51.6196i 1.79390i
829829 −46.3683 −1.61044 −0.805219 0.592978i 0.797952π-0.797952\pi
−0.805219 + 0.592978i 0.797952π0.797952\pi
830830 20.6023i 0.715116i
831831 −3.40835 −0.118234
832832 −47.3958 −1.64315
833833 0 0
834834 55.9460 1.93725
835835 −3.98742 −0.137990
836836 40.9676i 1.41690i
837837 4.47251 0.154593
838838 39.0882i 1.35028i
839839 24.8903i 0.859309i 0.902993 + 0.429654i 0.141365π0.141365\pi
−0.902993 + 0.429654i 0.858635π0.858635\pi
840840 − 0.554142i − 0.0191197i
841841 −16.9908 −0.585890
842842 −4.26559 −0.147002
843843 40.0176i 1.37828i
844844 − 16.4233i − 0.565314i
845845 8.97524i 0.308758i
846846 11.5285 0.396357
847847 − 1.75607i − 0.0603391i
848848 −22.7804 −0.782281
849849 74.0357 2.54090
850850 0 0
851851 −12.9947 −0.445453
852852 −9.92589 −0.340055
853853 − 28.1063i − 0.962342i −0.876627 0.481171i 0.840212π-0.840212\pi
0.876627 0.481171i 0.159788π-0.159788\pi
854854 −5.25850 −0.179942
855855 − 24.1455i − 0.825758i
856856 1.69749i 0.0580189i
857857 − 24.5101i − 0.837250i −0.908159 0.418625i 0.862512π-0.862512\pi
0.908159 0.418625i 0.137488π-0.137488\pi
858858 61.1889 2.08896
859859 12.8979 0.440069 0.220034 0.975492i 0.429383π-0.429383\pi
0.220034 + 0.975492i 0.429383π0.429383\pi
860860 26.7871i 0.913432i
861861 2.19704i 0.0748751i
862862 − 7.71840i − 0.262890i
863863 5.83668 0.198683 0.0993414 0.995053i 0.468326π-0.468326\pi
0.0993414 + 0.995053i 0.468326π0.468326\pi
864864 − 7.52970i − 0.256166i
865865 14.6078 0.496681
866866 −57.9014 −1.96757
867867 0 0
868868 4.06629 0.138019
869869 13.4538 0.456390
870870 35.4349i 1.20136i
871871 −8.78854 −0.297788
872872 5.62357i 0.190438i
873873 42.9266i 1.45285i
874874 99.4698i 3.36461i
875875 −0.368961 −0.0124732
876876 1.65266 0.0558382
877877 − 18.7846i − 0.634312i −0.948373 0.317156i 0.897272π-0.897272\pi
0.948373 0.317156i 0.102728π-0.102728\pi
878878 − 25.9746i − 0.876600i
879879 52.1040i 1.75742i
880880 8.34991 0.281475
881881 − 45.7423i − 1.54110i −0.637380 0.770549i 0.719982π-0.719982\pi
0.637380 0.770549i 0.280018π-0.280018\pi
882882 47.8655 1.61171
883883 6.15932 0.207277 0.103639 0.994615i 0.466951π-0.466951\pi
0.103639 + 0.994615i 0.466951π0.466951\pi
884884 0 0
885885 −1.22229 −0.0410870
886886 41.3147 1.38800
887887 − 16.7476i − 0.562329i −0.959660 0.281165i 0.909279π-0.909279\pi
0.959660 0.281165i 0.0907207π-0.0907207\pi
888888 −2.91270 −0.0977440
889889 5.80432i 0.194671i
890890 − 8.91981i − 0.298993i
891891 − 19.3880i − 0.649522i
892892 8.99726 0.301251
893893 11.8522 0.396620
894894 − 58.1500i − 1.94483i
895895 − 14.7452i − 0.492878i
896896 − 1.73862i − 0.0580834i
897897 79.2637 2.64654
898898 − 41.7976i − 1.39480i
899899 −32.6741 −1.08974
900900 7.70378 0.256793
901901 0 0
902902 12.2059 0.406412
903903 10.9032 0.362835
904904 − 6.35579i − 0.211391i
905905 11.2615 0.374344
906906 84.0518i 2.79243i
907907 − 8.12836i − 0.269898i −0.990853 0.134949i 0.956913π-0.956913\pi
0.990853 0.134949i 0.0430870π-0.0430870\pi
908908 − 7.95992i − 0.264159i
909909 30.5988 1.01490
910910 −3.58135 −0.118721
911911 − 35.6471i − 1.18104i −0.807023 0.590520i 0.798923π-0.798923\pi
0.807023 0.590520i 0.201077π-0.201077\pi
912912 − 60.4714i − 2.00241i
913913 − 24.8557i − 0.822605i
914914 −78.1431 −2.58474
915915 − 17.3692i − 0.574207i
916916 −10.0913 −0.333427
917917 1.50964 0.0498528
918918 0 0
919919 −24.2398 −0.799596 −0.399798 0.916603i 0.630920π-0.630920\pi
−0.399798 + 0.916603i 0.630920π0.630920\pi
920920 −3.98799 −0.131480
921921 41.2428i 1.35900i
922922 −79.0435 −2.60316
923923 8.06102i 0.265332i
924924 5.32031i 0.175025i
925925 1.93935i 0.0637654i
926926 27.2246 0.894657
927927 −16.0920 −0.528531
928928 55.0085i 1.80574i
929929 33.4397i 1.09712i 0.836111 + 0.548560i 0.184824π0.184824\pi
−0.836111 + 0.548560i 0.815176π0.815176\pi
930930 25.1747i 0.825510i
931931 49.2097 1.61278
932932 35.8363i 1.17386i
933933 88.5989 2.90060
934934 −33.5786 −1.09873
935935 0 0
936936 9.39647 0.307133
937937 −25.6264 −0.837177 −0.418588 0.908176i 0.637475π-0.637475\pi
−0.418588 + 0.908176i 0.637475π0.637475\pi
938938 − 1.43228i − 0.0467657i
939939 −79.0181 −2.57866
940940 3.78153i 0.123340i
941941 − 53.4661i − 1.74295i −0.490443 0.871473i 0.663165π-0.663165\pi
0.490443 0.871473i 0.336835π-0.336835\pi
942942 − 13.7591i − 0.448296i
943943 15.8115 0.514892
944944 −1.61901 −0.0526944
945945 − 0.342503i − 0.0111416i
946946 − 60.5738i − 1.96943i
947947 11.7772i 0.382707i 0.981521 + 0.191353i 0.0612877π0.0612877\pi
−0.981521 + 0.191353i 0.938712π0.938712\pi
948948 31.0873 1.00967
949949 − 1.34216i − 0.0435683i
950950 14.8450 0.481636
951951 31.6687 1.02693
952952 0 0
953953 23.9735 0.776576 0.388288 0.921538i 0.373067π-0.373067\pi
0.388288 + 0.921538i 0.373067π0.373067\pi
954954 47.5272 1.53875
955955 − 21.4934i − 0.695510i
956956 −15.0769 −0.487621
957957 − 42.7506i − 1.38193i
958958 − 4.29650i − 0.138814i
959959 5.98248i 0.193184i
960960 25.5135 0.823445
961961 7.78674 0.251185
962962 18.8244i 0.606924i
963963 9.60545i 0.309531i
964964 10.0199i 0.322720i
965965 −1.42208 −0.0457783
966966 12.9178i 0.415622i
967967 −45.6312 −1.46740 −0.733700 0.679473i 0.762208π-0.762208\pi
−0.733700 + 0.679473i 0.762208π0.762208\pi
968968 −2.83272 −0.0910471
969969 0 0
970970 −26.3919 −0.847394
971971 58.9182 1.89078 0.945388 0.325946i 0.105683π-0.105683\pi
0.945388 + 0.325946i 0.105683π0.105683\pi
972972 − 51.1693i − 1.64125i
973973 3.95052 0.126648
974974 − 49.7486i − 1.59405i
975975 − 11.8294i − 0.378845i
976976 − 23.0067i − 0.736425i
977977 24.7610 0.792176 0.396088 0.918213i 0.370368π-0.370368\pi
0.396088 + 0.918213i 0.370368π0.370368\pi
978978 −99.7670 −3.19020
979979 10.7614i 0.343934i
980980 15.7007i 0.501540i
981981 31.8217i 1.01599i
982982 −46.3410 −1.47880
983983 9.89686i 0.315661i 0.987466 + 0.157830i 0.0504499π0.0504499\pi
−0.987466 + 0.157830i 0.949550π0.949550\pi
984984 3.54406 0.112981
985985 −17.4127 −0.554815
986986 0 0
987987 1.53920 0.0489934
988988 76.8772 2.44579
989989 − 78.4669i − 2.49510i
990990 −17.4206 −0.553664
991991 − 0.883820i − 0.0280755i −0.999901 0.0140377i 0.995532π-0.995532\pi
0.999901 0.0140377i 0.00446850π-0.00446850\pi
992992 39.0807i 1.24081i
993993 1.25530i 0.0398356i
994994 −1.31372 −0.0416687
995995 −17.8876 −0.567076
996996 − 57.4332i − 1.81984i
997997 29.0009i 0.918467i 0.888315 + 0.459234i 0.151876π0.151876\pi
−0.888315 + 0.459234i 0.848124π0.848124\pi
998998 − 7.65728i − 0.242387i
999999 −1.80028 −0.0569582
Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1445.2.d.g.866.4 12
17.2 even 8 85.2.e.a.81.2 yes 12
17.4 even 4 1445.2.a.o.1.5 6
17.8 even 8 85.2.e.a.21.5 12
17.13 even 4 1445.2.a.n.1.5 6
17.16 even 2 inner 1445.2.d.g.866.3 12
51.2 odd 8 765.2.k.b.676.5 12
51.8 odd 8 765.2.k.b.361.2 12
68.19 odd 8 1360.2.bt.d.81.1 12
68.59 odd 8 1360.2.bt.d.1041.1 12
85.2 odd 8 425.2.j.c.149.5 12
85.4 even 4 7225.2.a.z.1.2 6
85.8 odd 8 425.2.j.c.174.5 12
85.19 even 8 425.2.e.f.251.5 12
85.42 odd 8 425.2.j.b.174.2 12
85.53 odd 8 425.2.j.b.149.2 12
85.59 even 8 425.2.e.f.276.2 12
85.64 even 4 7225.2.a.bb.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.e.a.21.5 12 17.8 even 8
85.2.e.a.81.2 yes 12 17.2 even 8
425.2.e.f.251.5 12 85.19 even 8
425.2.e.f.276.2 12 85.59 even 8
425.2.j.b.149.2 12 85.53 odd 8
425.2.j.b.174.2 12 85.42 odd 8
425.2.j.c.149.5 12 85.2 odd 8
425.2.j.c.174.5 12 85.8 odd 8
765.2.k.b.361.2 12 51.8 odd 8
765.2.k.b.676.5 12 51.2 odd 8
1360.2.bt.d.81.1 12 68.19 odd 8
1360.2.bt.d.1041.1 12 68.59 odd 8
1445.2.a.n.1.5 6 17.13 even 4
1445.2.a.o.1.5 6 17.4 even 4
1445.2.d.g.866.3 12 17.16 even 2 inner
1445.2.d.g.866.4 12 1.1 even 1 trivial
7225.2.a.z.1.2 6 85.4 even 4
7225.2.a.bb.1.2 6 85.64 even 4